Systems, apparatuses, and methods for optical focusing in scattering samples

ABSTRACT

A method includes applying, to a sample exhibiting optical scattering and having a emission particles distributed therein that exhibit spin-dependent fluorescence, a magnetic field to shift a resonance frequency of each emission particle in a position-dependent manner. The method also includes exciting the sample with an excitation beam that causes at least one emission particle to emit spin-dependent fluorescence and detecting the emitted spin-dependent fluorescence. The method also includes estimating a position of the emission particle(s) within the sample based on the spin-dependent fluorescence, the resonance frequency, and the magnetic field. The method also includes estimating optical transmission information for the sample based on a wavefront of the excitation beam and the estimated position. The optical transmission information including a measure of an optical field at each position of an emission particle.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Application No.62/618,297 filed Jan. 17, 2018, titled “QUANTUM REFERENCE BEACON”, theentire disclosure of which is hereby incorporated by reference.

STATEMENT OF GOVERNMENT SUPPORT

This invention was made with government support under Grant No.134062-5093041 awarded by the Army Research Office. The government hascertain rights in the invention.

BACKGROUND

Optical random scattering is generally considered to be a nuisance ofmicroscopy that limits imaging depth and spatial resolution. Wavefrontshaping techniques have recently enabled optical imaging atunprecedented depth, but a remaining problem is also to attainsuper-resolution within complex, scattering media.

SUMMARY

In some embodiments, a method includes applying, to a sample exhibitingoptical scattering and having emission particles distributed thereinthat exhibit spin-dependent fluorescence, a magnetic field to shift aresonance frequency of each emission particle in a position-dependentmanner. The method also includes exciting the sample with an excitationbeam that causes at least one emission particle to emit spin-dependentfluorescence, and detecting the emitted spin-dependent fluorescence. Themethod also includes estimating a position of the emission particle(s)within the sample based on the spin-dependent fluorescence, theresonance frequency, and the magnetic field. The method also includesestimating optical transmission information for the sample based on awavefront of the excitation beam and the estimated position. The opticaltransmission information including a measure of an optical field at eachposition of an emission particle.

In some embodiments, a system includes a magnet to apply, to a sampleexhibiting optical scattering and having a plurality of emissionparticles distributed therein that exhibit spin-dependent fluorescence,a magnetic field to shift a resonance frequency of each emissionparticle of the plurality of emission particles in a position-dependentmanner. The system also includes a microwave generator to generate andapply a microwave signal to the sample to modulate the spin state of atleast one emission particle of the plurality of emission particles. Thesystem also includes a light source to excite the sample with anexcitation beam, such that the excitation beam causes at least oneemission particle of the plurality of emission particles to emitspin-dependent fluorescence. The system also includes a detector todetect the spin-dependent fluorescence. The system further includes acontroller to estimate a position of the at least one of the pluralityof emission particles within the sample based on the spin-dependentfluorescence, the resonance frequency of the at least one of theplurality of emission particles, and the magnetic field. The controlleralso estimates optical transmission information for the sample based ona wavefront of the excitation beam and the estimated position. Theoptical transmission information includes a measure of an optical fieldat each position of an emission particle of the plurality of emissionparticles.

In some embodiments, a system includes a magnet, in electromagneticcommunication with a sample exhibiting optical scattering and having aplurality of diamond nanoparticles distributed therein with nitrogenvacancy centers that exhibit spin-dependent fluorescence. The magnetapplies a magnetic field having a gradient, such that the magnetic fieldshifts a resonance frequency of each nitrogen vacancy center in theplurality of diamond nanoparticles based on a position of that nitrogenvacancy center in the sample. The system also includes a light source toemit an excitation beam, and a wavefront shaping device, in opticalcommunication with the light source and the sample, to excite the samplewith the excitation beam. The excitation beam includes a set of basismodes, and the excitation beam causes at least one nitrogen vacancycenter in the plurality of diamond nanoparticles to emit spin-dependentfluorescence. The system also includes a detector, in opticalcommunication with the sample, to detect the spin-dependent fluorescenceemitted by the sample. The system also includes a controller operablycoupled to the detector and the wavefront shaping device. The controllerestimate a position of the at least one nitrogen vacancy center in theplurality of diamond nanoparticles within the sample based on thespin-dependent fluorescence, the resonance frequency of the at least onenitrogen vacancy center in the plurality of diamond nanoparticles, andthe magnetic field. The controller also generates an opticaltransmission matrix for the sample based on a wavefront of theexcitation beam and the estimated position. The optical transmissionmatrix includes a measure of an optical field at a position of eachnitrogen vacancy center in the plurality of diamond nanoparticles, andfor each basis mode of the set of basis modes. The controller alsoestimates, for the position of each nitrogen vacancy center in theplurality of diamond nanoparticles and for each basis mode, a feedbackparameter to generate a set of feedback parameters. The controller alsoactuates the wavefront shaping device to modify at least onecharacteristic of at least one basis mode of the excitation beam basedon the set of feedback parameters.

It should be appreciated that all combinations of the foregoing conceptsand additional concepts discussed in greater detail below (provided suchconcepts are not mutually inconsistent) are contemplated as being partof the inventive subject matter disclosed herein. In particular, allcombinations of claimed subject matter appearing at the end of thisdisclosure are contemplated as being part of the inventive subjectmatter disclosed herein. It should also be appreciated that terminologyexplicitly employed herein that also may appear in any disclosureincorporated by reference should be accorded a meaning most consistentwith the particular concepts disclosed herein.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawing(s) will be provided by the Office upon request and paymentof the necessary fee.

The skilled artisan will understand that the drawings primarily are forillustrative purposes and are not intended to limit the scope of theinventive subject matter described herein. The drawings are notnecessarily to scale; in some instances, various aspects of theinventive subject matter disclosed herein may be shown exaggerated orenlarged in the drawings to facilitate an understanding of differentfeatures. In the drawings, like reference characters generally refer tolike features (e.g., functionally similar and/or structurally similarelements).

FIG. 1A illustrates how optical random scattering in complex mediadistorts the incident optical field. However, this distortion can bereversed by shaping the incident wavefront. Embedded quantum referencebeacons (QRBs) provide feedback about subwavelength features of thescattered optical fields, guiding the wavefront shaping process. Thisapproach enables, for example, super-resolution focusing deep inside ofcomplex media or individual spin-qubit measurement in adiffraction-limited area (the dashed circle).

FIG. 1B illustrates nitrogen-vacancy (NV) centers in diamond withspin-dependent fluorescence. Electrons with the spin magnetic sublevels|m_(s)=±1

preferentially decay (dashed arrow) to the dark metastable state (¹A),once they are optically pumped to the excited states ³E (arrow),resulting in reduced fluorescence than that from the sublevel |m_(s)=0

. This spin-dependent fluorescence enables optically detectable magneticresonance (ODMR).

FIG. 1C illustrates the QRB-guide star (GS) feedback that is producedwith the spin-dependent fluorescence. To measure the optical field onthe QRB positioned at x₁, its fluorescence is selectively reduced byelectron spin resonance (ESR). The change of collected fluorescencedetermines the optical field at x₁. This process can be repeated foranother position at x₂ as shown in the bottom plot.

FIG. 2A shows the different resonant frequencies of the QRBs based onfluorescence collection, where {ρ_(i)} labels the electron spin statesof QRBs, and an external magnetic field gradient splits their individualresonance frequencies. Quantum operators {E_(i)} drive the electron spintransition of target QRBs.

FIG. 2B illustrates measurement sequences for the iterative wavefrontoptimization. The Fourier basis modes of the incident wavefront (k₁, k₂,. . . ) are encoded into holographic illuminations, in which the basismodes interfere with the reference plane wave for complex field readout.The overall fluorescence difference with {E_(i)} (i.e.,N_(j)[ρ]-N_(j)[E_(i)(ρ)]) produces the QRB-GS feedback S_(i,j). ϕdescribes the phase of each basis mode relative to the reference planewave.

FIG. 2C illustrates modulation of the QRB-GS feedback in the iterativewavefront optimization. In each step, the phase ϕ of the basis modes isadjusted to compensate for the phase offset of the modulation.

FIGS. 2D, 2F illustrate iterative wavefront optimization with the QRB-GSfeedback. Two QRBs have the electron spin resonance frequencies atv₁=2.825 GHz and v₂=2.762 GHz. The resonant microwaves continuouslydrive the resonances to produce the QRB-GS feedback, so that theincident optical fields can be iteratively updated to optimize theQRB-GS feedback signal strength at v₁ (FIG. 2D) and v₂ (FIG. 2F).

FIGS. 2E, 2G illustrate the ODMR contrast at v₁ (FIG. 2E) and v₂ (FIG.2G) for the iterative optimization processes.

FIG. 3A illustrates NV centers in subwavelength nanodiamonds (DiamondNanotechnologies) embedded in a complex medium including randomlydistributed TiO2 nanoparticles (Sigma Aldrich 718467). A green laserbeam is delivered to the complex medium by a microscope objective (0.8NA, 60x). An objective (0.95 NA, 100x) at the other side directlycollects spin-dependent broadband red fluorescence from NV centers. Thethickness of the complex medium is ˜7±2 μm.

FIG. 3B illustrates an experimental setup. A digital micromirror device(DMD) shapes the wavefront of the incident green laser and projects itonto the back aperture of excitation objective. The phase of eachincident basis mode {k_(n)} is controlled by groups of 24 by 24 DMDmicromirrors. SPCM (CCD) counts (images) the red fluorescence collectedby the collection objective. LP rejects the transmitted green laser, anda pinhole in front of SPCM blocks stray red fluorescence. A copper wire(diameter of 25 μm) delivers the microwave signal, either directly orafter amplification, to QRBs to modulate their spin ground statepopulation. A permanent magnet separates the magnetic resonancefrequencies of the QRBs by orientation-dependent Zeeman splitting. DMD:a digital micro-mirror device, SPCM: a single photon counting module,CCD: a charge-coupled device, LP: a long-pass optical filter with acutoff wavelength of 650 nm, and L1-L6: lens.

FIG. 3C illustrates QRB fluorescence images for super-resolutionfocusing demonstration. x₁ and x₂ denote the QRB positions. Inset images(large panel−fluorescence image of both QRBs without ESR, smallerpanels−fluorescence images of individuals QRBs with ESR) are obtainedusing super-resolution focusing of the QRB-assisted wavefront shapingtechnique. All scale bars=0.61λ/NA with NA=0.8 and λ=532 nm.

FIG. 3D, top panel illustrates how an external magnetic field gradientinduces an energy shift in ground spin state in two QRBs at differentpositions based on the Zeeman effect. The bottom panel illustrates howthe energy shift, and corresponding magnetic resonance, can be opticallydetected by spin-dependent fluorescence and application of a microwavesignal.

FIG. 4A illustrates the phase-only wavefront W_(v1) determined byoptimizing the QRB-GS feedback at v₁.

FIG. 4B illustrates the phase-only wavefront W_(v2), determined byoptimizing the QRB-GS feedback at v₂.

FIG. 4C illustrates the phase-only wavefront W_(c1), obtained using thefluorescence GS method.

FIG. 4D illustrates ODMR spectra with W_(v1), W_(v2), and W_(c1)projection.

FIG. 4E illustrates spatial resolution of the subwavelength foci in thecomplex medium. Two lines plot the estimated intensity shape of thesubwavelength foci with W_(v1) and W_(v2) projection, respectively. Theshaded area gives the estimation uncertainty. The other solid line plotsthe point spread function (PSF) of the excitation objective(FWHM=0.61λ/NA with λ=532 nm and NA=0.8), and the dashed line refers tothe far-field limited PSF (NA=1). Inset: reconstructed image of thesubwavelength foci with W_(v1) and W_(v2). Scale bar =0.61λ/NA withNA=0.8 and λ=532 nm.

FIG. 5A illustrates modulating Δσ_(i) of a target QRB withcontinuously-driven ESR. For a given holographic illumination, themicrowave is continuously applied at the reference frequency ofv_(ref)=2.5 GHz, which is far off from v₁=2.825 GHz and v₂=2.762 GHz,for 20 ms, and at the target resonance frequency v₁ or v₂ for another 20ms. During the microwave operations, the spin-dependent fluorescencephoton N[ρ] and N[E(ρ)] are simultaneously collected with a singlephoton counting module. This unit sequence is repeated for 300 times pera holographic illumination. A digital clock pulse train from a DAQsynchronizes the microwave operations and the photon collections.

FIG. 5B illustrates how the DMD in the optical microscope projects theholographic illumination 1+e^(ik.r) of the incident basis mode. Signalgenerator applies the quantum operator E to produce Δσ_(i). SPCM countsthe spin-dependent fluorescence photons, and DAQ returns thefluorescence measurement N[ρ], N[E(ρ)]. From the measurements, personalcomputer determines the phase ϕ to be compensated on the incident basismode. Updating the phase ϕ to DMD closes the iterative wavefrontoptimization cycle. DMD: digital micromirror device (D4100, DigitalInnovations), SPCM: single photon counting module (SPCM-AQ4C,Excelitas), DAQ: Multi-functional data acquisition (NI-6343, NationalInstrument), Signal Generator (SME Rohde & Schwarz).

FIG. 6A illustrates the phase maps of 793 incident basis modes,determined by the QRB-GS feedback. The DMD projects the phase maps intothe back aperture of the excitation objective lens. Left (Right) plot isthe result of the iterative optimization with the QRB-GS feedback at v₁(v₂).

FIGS. 6B, 6C illustrate the intensity (FIG. 6B) and phase map (FIG. 6C)of the incident wavefront on the complex medium, respectively. Thesemaps are obtained by the Fourier transform of the phase maps plotted inFIG. 6A. (Scale bars=3 μm.)

FIGS. 7A, 7B illustrate individual excitation of QRB₁ (FIG. 7A) and QRB₂(FIG. 7B) with subwavelength optical focus through a scattering medium.The central position x₁ and x₂ of QRB1 and QRB2 are localized by fittingthe recorded fluorescence images into two-dimensional Gaussianfunctions. (Scale bar=1.22λ/NA≈810 nm.)

FIG. 7C illustrates the central position with the W_(c1) projection forcomparison.

FIG. 7D illustrates merged positions from FIGS. 7A-7D. The dashed circleguides the diffraction-limited resolution of the excitation microscopeobjective (NA=0.8, λ=532 nm).

FIG. 8 is a flow chart of a method for optical focusing, according toembodiments.

FIG. 9 is a system for optical focusing, according to embodiments.

FIG. 10 illustrates an example setup with a super-lens coupled to asample.

DETAILED DESCRIPTION

The concepts introduced above and discussed in greater detail below maybe implemented in numerous ways. Examples of specific implementationsand applications are provided primarily for illustrative purposes toenable those skilled in the art to practice the implementations andalternatives apparent to those skilled in the art.

The figures and example implementations described below are not meant tolimit the scope of the present implementations to a single embodiment.Other implementations are possible by way of interchange of some or allof the described or illustrated elements. Moreover, where certainelements of the disclosed example implementations may be partially orfully implemented using known components, in some instances only thoseportions of such known components that are necessary for anunderstanding of the present implementations are described, and detaileddescriptions of other portions of such known components are omitted soas not to obscure the present implementations.

Following below are more detailed descriptions of various conceptsrelated to, and implementations of, optical focusing in scatteringsamples. In particular, disclosed herein is an example technique tofocus inside of complex, scattering media by the use of a quantumreference beacon (QRB), which can include solid-state quantum emitterswith spin-dependent fluorescence. This QRB provides subwavelength“guidestar” (described in more detail herein) feedback for wavefrontshaping to achieve an optical focus below a microscope's diffractionlimit. The QRB-guided imaging approach is implemented usingnitrogen-vacancy centers in diamond nanocrystals (also sometimesreferred to as nano-diamonds, diamond nanoparticles, nano-diamondparticles, and variants thereof), which enable optical focusing with asub-diffraction resolution below 186 nm (≈λ/3.5 NA), where themicroscope's NA=0.8. This QRB-assisted wavefront shaping paves the wayfor a range of applications, including deep-tissue quantum enhancedsensing, and individual optical excitation of magnetically-coupled spinensembles for applications in quantum information processing.

Optical random scattering in complex media, such as biological tissues,distorts an incident optical focus, reducing the resolution and imagingdepth of optical microscopy. However, random scattering does not lead tothe permanent loss of focusing capability; instead it randomizes theincident focus in a deterministic way. By reversing this scattering, itbecomes possible to focus and even to image through complex media.Moreover, random scattering can actually benefit microscopy bypermitting a spatial resolution below the diffraction-limit of λ/2 NA,where NA is the numerical aperture of the microscope objective. Thissuper-resolution is possible because random scattering couples opticalmodes with high in-plane momentum from the sample to the microscopeobjective, much like a disordered grating. By extending this principleto evanescent modes of the sample, far-field superlenses for near-fieldfocusing and imaging have been achieved.

Reversing random scattering uses feedback from the target focal points.In particular, focusing light inside of complex media can use a type of“guidestar (GS)” that provides feedback of the interior optical field.This feedback guides incident wavefront adjustments to focus thescattered light into the GS point. In the last decade, various forms ofGSs have been implemented, including fluorescence, ultrasound, nonlinearreference beacons, and kinetic objects. However, the spatial resolutionusing these types of GSs has been far from the super-resolution limit.To push this resolution to or below the diffraction limit can includetwo key advances: (i) the physical size of the GS should be ofsubwavelength scale, and (ii) it should be possible to resolvesub-diffraction features of randomly scattered light. A subwavelengthaperture used in scanning near-field optical microscopy (SNOM) satisfiesthese conditions, but this technique does not permit imaging within acomplex medium. To address these challenges, quantum reference beacons(QRBs) are introduced herein.

The QRB proposed herein includes solid-state quantum emitters withspin-dependent fluorescence. An example is the nitrogen vacancy (NV)center in diamond, which has emerged as a leading quantum system forquantum sensing and quantum information processing. By resonantlydriving electron spin transitions of each QRB, the spin-dependentfluorescence produces the subwavelength GS feedback that enablessuper-resolution focusing within complex media. This proposal isdemonstrated herein with ensembles of NV centers in subwavelengthdiamond nanocrystals, and show super-resolution focusing inside of adisordered scattering medium with a resolution below 186 nm (≈λ/3.5 NAwith λ=532 nm).

FIGS. 1A-1C illustrate the approach to QRB-guided wavefront shaping inmicroscopy. A wavefront shaper adjusts basis modes (shown as individualpixels in FIG. 1A) of the incident wavefront to interfere scatteredlight constructively at target GS points. This specific wavefrontadjustment is determined from the QRB-GS feedback. This feedback signalis created by applying a magnetic field gradient across the sample sothat one of several QRBs inside a diffraction-limited volume can beselectively driven into its dark magnetic sublevels, as indicated inFIG. 1C and detailed below.

Specifically, the QRB-GS feedback signal is used to measure thetransmission matrix that characterizes the light propagation through acomplex medium, as discussed in more detail herein. The electron spinstate of the embedded QRBs is labeled at {x_(i)}=x₁, . . . , x_(N) witha spin density operator ρ=ρ₁⊗ρ₂⊗ . . . ⊗P_(N). An external magneticfield gradient separates their resonant frequencies {v_(i)} by theZeeman effect as shown in FIG. 1A. In principle, {x_(i)} could then bereconstructed from {v_(i)} and knowledge of the external magnetic fieldgradient. Resonant driving of each {ρ_(i)} spin transition isrepresented through a quantum operator {E_(i)}. When the jth incidentbasis mode is coupled into the medium, the QRB-GS feedback S_(i,j) forx_(i) is described by

S _(i,j) =N _(j)[ρ]−N _(j)[E _(i)(ρ)]=|t _(i,j)|²Δσ_(i)Δ_(γ).  (1)

Here, N_(j)[ρ] and N_(j)[E_(i)(ρ)] denote the fluorescence photonnumbers collected for unit integration, t_(i,j) is the transmissionmatrix element (i.e., the scattered optical field at x_(i) for the jthincident basis mode), Δσ_(i)=½ tr[σ_(z){ρ_(i)−E_(i)(ρ_(i))}] where σ_(z)is the Pauli-z operator, and Δγ represents the variance of the collectedspin-dependent fluorescence between the optically bright and dark spinstates (FIG. 1B). FIGS. 2A-2G summarize the iterative wavefrontadjustments due to the QRB-GS feedback.

The spatial resolution of the methods described herein can be determinedby the ESR lineshape as detailed below, since the lineshape sets thepoint spread function (PSF) of the QRB-GS feedback that confines {E_(i)}only to the target QRBs (FIG. 2A). Specifically, a magnetic fieldgradient dB/dx translates the (mean) resonance linewidth δv to thespatial resolution Δd_(QRB) of the effective PSF:

$\begin{matrix}{{\Delta \; d_{QRB}} = \frac{\delta \; v}{\gamma_{e}\left( {{dB}/{dx}} \right)}} & (2)\end{matrix}$

where γ_(e) is the gyromagnetic ratio of the electronic spin (˜2.8MHz/Gauss). Combined with the crystal orientation-dependent Zeemansplitting and dynamical decoupling to narrow the linewidth, thisresolution can go down to a few tens of nanometers. Generally, themagnetic field gradient dB/dx can be based on the desired spatialresolution and the spin quality of QRB (i.e., linewidth of the magneticresonance). Alternatively, if the QRBs within a diffraction-limited areahave different orientation, orientation-dependent Zeeman splittingpermits for splitting the resonance frequencies with a constant magneticfield.

FIGS. 3A-3C illustrate an experimental configuration/system fordemonstrating QRB-assisted wavefront shaping. Generally, FIG. 3Aillustrates a system 300 for optical focusing that includes a sample 310having nanodiamonds 315 embedded therein (as best illustrated in FIG.3A. While the sample 310 is illustrated herein as being an engineeredsample that has randomly distributed TiO₂ nanoparticles to affectoptical scattering, it is understood that any scattering sample can beused with the system 300, including naturally occurring materials,engineered materials, biological materials such as tissues, andcombinations thereof. Additionally, while each QRB is illustrated hereinas a nanodiamond (e.g, a diamond nanocrystal having a nitrogen vacancycolor center), it can be any other suitable solid-state quantum emitterthat exhibits spin-dependent fluorescence.

The system 300 includes a magnet 320 that applies a magnetic field tothe sample 310 and establishes a gradient that shifts the resonantfrequency of each nanodiamond 315 based on its position within thesample 310. The magnet 320 can be a permanent magnet (e.g., having agrade of N42 in some cases) having a fixed/constant vector magneticfield, or an electromagnet with a variable/tunable magnetic field. Themagnet 320 can encompass a set of magnets, such that each magnetgenerates its own magnetic field along a different axis of the sample.In other instances, the magnet 320 can be movable around the sample toapply its magnetic field along different axes of the sample. Forexample, the magnet 320 can encompass multiple electromagnets alignedthe X, Y, Z axes of the sample, with each electromagnet independentlyswitched on and off to permit sequential application along the differentaxes.

The system 300 also includes a laser light source 340 to generate anexcitation beam that can be applied to the sample 310. While illustratedherein as a laser operating at a wavelength of about 532 nm, the lightsource 340 can be any suitable source to induce fluorescence emission inthe sample 310 (e.g., a laser, a lamp, light emitting diodes), andoperating at any suitable wavelength/range, including between 200 nm-700nm. The light source 340 is coupled to a wavefront shaping device, suchas the digital micromirror device (DMD) 350, though it is understoodthat other suitable spatial light modulators, such as a magneto-opticspatial light modulators, can also be employed. The DMD 350 can be usedto affect, at a per basis mode/per pixel level, modulation of one ormore characteristics (e.g., phase, amplitude) of the excitation beam foriterative wavefront shaping as discussed in more detail herein. Anexcitation objective 312 a (shown here as a 60x, 0.8 NA objective)couples the excitation beam to the sample 310, and an emission objective312 b (shown here as a 100x, 0.95 NA objective) receives the emissionfrom the sample 310. It is understood that other specifications ofmagnification and numerical aperture are possible and independentlyselectable for the objectives 312 a, 312 b. Aspects of the system 300are useful for super-resolution focusing, where the nanodiamonds 315 areseparately detectable even when their separation is lower than thediffraction limit of the excitation objective 312 a.

The excitation beam can include a reference part, or portion, or mode,and also include a set of spatial modes (also sometimes referred to as“basis modes”). The reference part can be coupled to the excitationobjective 312 a via a different optical path than the set of spatialmodes. For example, in an example setup, a beam splitter splits theexcitation beam into two separate beams, one of which is sent directlyto the same as the reference beam, while the other is sent to the samplevia scanning mirrors, an acousto-optic deflector, etc. as the set ofspatial modes. It is the set of spatial modes that undergo optimization(e.g., of phase) based on the QRB-GS feedback. The resultingfluorescence, absorbance, and/or transmission resulting from thereference part provides an additional or alternative approach toestimating the optical transmission information.

The fluorescence emission can then be captured by a detector 360, shownhere as a SPCM, which can include a photocathode, photoanode, aCCD-based sensor, a CMOS-based sensor, etc. The system 370 also includesan imaging device 370, shown here as a CCD camera, for capturingfluorescence images of the sample 310. During operation, thefluorescence signal, which represents spin-dependent fluorescenceemanating from the nanodiamonds 315, can be used to estimate thelocation of the nanodiamonds 315, and which in turn can be used toestimate optical transmission (i.e., including scattering)characteristics of the sample 310. This optical transmissioninformation, based on fluorescence from all the nanodiamonds 315 can beemployed to focus the interior optical field and generate the wavefrontW_(c1) as explained herein.

To obtain super-resolution as described herein, the system 300 alsoincludes a microwave generator 330 to generate a microwave signal havingstable amplitude and phase, and an amplifier 335 (coupled thereto,though the amplifier, which can be a low-noise amplifier, can beoptional. Generating the feedback signal (also sometimes referred to asa feedback parameter, or a set of feedback parameters) is done byapplying the microwave signal to the sample 310 to selectively drive thespin transition of each nanodiamond 315 separately and sequentially. Forexample, the microwave signal can have a frequency that is resonant withjust a first nanodiamond, resulting in a fluorescence response just fromthat nanodiamond, and (when provided as feedback to the DMD 350)resulting in a wavefront that is optimized for that nanodiamond. Then,the microwave signal can be reapplied at a frequency that is resonantwith that of a second nanodiamond, and that fluorescenceresponse/resulting wavefront is optimized for that nanodiamond. FIG. 3Cillustrates the fluorescence images of a) two such nanodiamonds whenimaged using the light source 330 (large panel), and b) the samenanodiamonds when the microwave generator 330 and the DMD 350 are alsoemployed as described herein, providing two separate fluorescenceimages, one or each nanodiamond (two smaller panels). In someembodiments, as described in greater detail with respect to FIGS. 5A-5B,the system 300 can encompass timing operations that coordinate operationof the microwave source 330 and the light source 340 (and/or the DMD350) for fluorescence image collection for a particular nanodiamond.

In the specific configuration illustrated in FIGS. 3A-3C, the QRBs areensembles of NV centers (FIG. 1B) in nanodiamonds with a mean diameterof 50 nm. The QRBs are embedded in a complex medium made of randomlydistributed TiO₂ nanoparticles with a mean diameter of 21 nm. Theincident green laser light (λ=532 nm) is randomly scattered as itpropagates through the medium. This scattering produces subwavelengthspatial features on the incident laser light, which excite the embeddedQRBs. In particular, demonstrated herein is super-resolution focusing ontwo QRBs at x₁ (QRB₁) and x₂ (QRB₂) in FIG. 3C, where their separation|x₁−x₂=186 nm is far below the diffraction limit of the excitationobjective lens, 406 nm (FIGS. 7A-7D). The QRB₁ (QRB₂) has an ESRfrequency of v₁=2.825 GHz (v₂=2.762 GHz), which corresponds to theelectronic spin transition between |m_(s)=0

and one of the Zeeman split |m_(s)=±1

of the ground spin triplet (³A, FIG. 1B). Since v₁ and v₂ arewell-separated (Δv≈63 MHz) compared to their resonance linewidths (δv₁=5MHz and δv₂=5.6 MHz), it is possible to individually drive the spintransition of each QRB.

In this study, the incident wavefront is shaped with 793 transverseFourier basis modes {k_(n)}, which cover the entire back aperture of theexcitation objective. Resonant microwaves drive the spin transitions atv₁ and v₂ that produce the QRB-GS feedback, and the phase of {k_(n)} isiteratively adjusted to optimize the feedback signal (FIGS. 2D and 2F).FIGS. 4A, 4B plot the results of the wavefront optimizations W_(v1) andW_(v2), respectively. For comparison, FIG. 4C shows the wavefrontW_(c1), obtained without the use of ESR (i.e. by optimizing fluorescencefeedback from QRBs). This fluorescence GS method focuses the interioroptical field without achieving super-resolution.

Projecting the wavefronts W_(v1) (W_(v2)) forms a superresolutionoptical focus at x₁(x₂) in the complex medium. This super-resolutionfocusing can be verified by investigatingoptically-detectable-magnetic-resonance (ODMR) spectra. This is because(i) ODMR spectra exhibit resonances of optically pumped QRBs, and (ii)QRB1 and QRB2 have distinguishable spectra. FIG. 4D plots the ODMRspectra for this investigation. First, the wavefront W_(c1) is projectedwith the DMD (FIG. 3B), which produces the ODMR spectrum as shown. Thisspectrum shows the resonances at v₁ and v₂ of both QRBs, as expected. Bycontrast, the resonance of QRB₁ appears when W_(v1) is projected, whichis obtained using the QRB-GS feedback with the spin transition at v₁.Alternatively, projecting W_(v2) reveals the resonance of QRB₂. Thisdemonstration validates the ability of QRB-guided wavefront shaping toenable optical addressing of individual spots far below the diffractionlimit. Note that the resonance linewidths are slightly broadened whenthe QRBs are excited by the targeted subwavelength foci, owing to theoptically induced relaxation of ODMR.

The ODMR spectra with subwavelength spin addressing enable us toestimate the spatial resolutions of the optical foci (FIG. 4E). Thepeak-to-background intensity ratio of the focus (i.e. I(x₁)=I(x₂) orvice versa) is determined from the ODMR spectra, as detailed herein.Assuming the subwavelength focus features a Gaussian intensity envelope,the intensity ratio indicates that the super-resolution focus at x₁ (x₂)has a spatial resolution of 204 nm (184 nm). This achieved resolution is2 (2.21) times smaller than the diffraction limited resolution and 1.31(1.45) times smaller than the far-field-limited one (NA=1).

FIG. 5A illustrates timing aspects of microwave generation via themicrowave generator 530 (referred to here as a signal generator) andfluorescence collection via the SPCM 560. The sample is illuminatedcontinuously for a 40 ms duration via excitation beam delivery from theDMD 550. During the first 20 seconds, the microwave generator 530applies a frequency (v_(ref)) that is different than the resonantfrequency of all nanodiamonds, and the SPCM 560 collects fluorescencephotons N[ρ] that represent fluorescence from all nanodiamonds as areference. During the next 20 seconds, the microwave generator 530applies a frequency (v_(i)) that is the same as the resonant frequencyof a target nanodiamond, and the SPCM 560 collects fluorescence photonsN[E_(i)(ρ)] that represents fluorescence from that target nanodiamond. Atiming device, such as a digital acquisition (DAQ) circuit, can be usedto control operation of the DMD 550, the signal generator 530, and theSPCM 560. The measurement can be repeated for each nanodiamond toimprove signal-to-noise (SNR) of the measurement (e.g., repeated 10, 50,100, 300, 500 times, etc.).

FIG. 5B illustrates how the fluorescence collection in FIG. 5A is usedto iteratively modify the wavefront of the excitation beam/theholographic illumination, denoted here by 1+e^(ik.r). The holographicillumination of each basis mode is applied to the sample (containedwithin the microscope 575) via the DMD 550, and the signal/microwavegenerator applies the microwave signal E_(i). The fluorescencemeasurements N[ρ], N[E(ρ)] are used to update (by a personal computer580, which can include a controller 980 as disclosed in FIG. 9) thephase ϕ of that hologram of the corresponding basis mode. This entireloop can be run iteratively until the QRB-GS signal, undergoingoptimization, reaches a predetermined criterion, such as saturation forexample.

These data show that a QRB enables super-resolution optical focusingwithin complex media. This QRB-GS approach uniquely provides, for thefirst time, sub-wavelength guidestar feedback inside a scattering mediumby the use of spin coherence. Implementing this proposed approach withNV centers demonstrates clear super-resolution focusing capabilitiesinside of a complex medium. This QRB-assisted wavefront shaping opens upa range of applications. First, it can extend to quantum sensing basedon NV centers to greater imaging depth and optical super-resolution.Second, it can be used to characterize the light propagation through afiber for single-fiber endomicroscopy. Finally, our method could open upthe way for subwavelength optical spin measurement of magneticdipole-coupled quantum emitters, which is essential for advanced quantumsensing, quantum error correction, and room-temperature quantumcomputing.

Electronic Spin Resonance of Quantum Reference Beacons. The quantumreference beacon (QRB) includes a solid-state quantum emitter withspin-dependent fluorescence. One example is a nitrogen-vacancy (NV)center in diamond, which has the electron spin magnetic sublevels|m_(s)=±1

that are optically darker than |m_(s)=0

. This spin-dependent fluorescence enables optical detection of magneticresonance. The QRB-guidestar feedback is based on the opticallydetectable magnetic resonance (ODMR), in which the optical fields areread out at the resonance points inside of complex media. By driving theresonances of the QRBs, their spin populations are individuallymodulated below the optical diffraction-limited resolution, whichproduces the guidestar feedback for wavefront shaping. In the NV-basedexperiment, the resonance is driven between the bright |m_(s)=0

and one of the dark |m_(s)=±1

of NV centers. In the following, the bright and dark spin state involvedin the magnetic resonance are represented as |0

and |1

, respectively, and denote a spin state of a QRB with a spin densityoperator ρ.

For QRB-assisted wavefront shaping, the evolution of ρ is described bythe master equation

$\begin{matrix}{\frac{dp}{dt} = {{\frac{1}{i\; \hslash}\left\lbrack {H,\rho} \right\rbrack} + \left\{ \frac{d\; \rho}{dt} \right\}_{realx}}} & (3)\end{matrix}$

where H is a simple two-level spin Hamiltonian that describes therelevant interaction of QRB with a microwave:

H=hω|1

1|+hΩ cos(ω_(mw))(|0

1|+|1

0|).

Here, the energy splitting ℏω includes the zero-field splittingD_(gs)≈2.87 GHz and the electronic Zeeman splitting γ_(e)B_(0z), whereγ_(e)=2.8 MHz /Gauss and B_(0z) is a magnetic field along the symmetryaxis of NV centers. ω_(mw) comes is a microwave frequency that drivesthe magnetic resonance with the Rabi frequency Ω.

The last term of the master equation Eq. (3) represents the spinrelaxation due to the interaction with the QRB's environment. Thisrelaxation process includes the intrinsic spin relaxation from magneticdipolar interactions with a spin bath. In addition, optical excitationinduces spin relaxation through (i) spin polarization via intersystemcrossing (ISC) followed by non-radiative decay, and (ii) the decoherencewith scattered photons. Typical values of the intrinsic and opticallyinduced relaxation rates for NV centers can be found in A. Dreau et al.

Modulating the spin state ρ with a resonant microwave produces theguidestar feedback (the QRB-GS feedback). The modulation is denotedthrough a quantum operator E that maps an initial spin state ρ to themodulated state E(ρ). In this experiment, ρ is modulated by continuousESR (electronic spin resonance) spectroscopy, in which ρ and E(ρ) arethe steady-steady solutions of Eq. (3) under optical excitation. Withthe modulation, the change of the spin population Δσ is

Δσ=½tr[σ_(z){ρ−E(ρ)}],  (4)

where σ_(z) is the Pauli-z operator, and tr[•] is the trace operator.For example, Δσ=½ for continuously-driven ESR of an initially polarizedspin (i.e. ρ=|0

0|), and Δσ=1 for an ideal spin flip with a microwave π-pulse. For thecase that the microwave has a detuning δ from the spin resonancefrequency, Δσ is reduced by an ESR lineshape g(δ) with g(0)=1:

½tr[σ_(z){ρδ−E(ρδ)}]=g(δ)Δσ.  (5)

Here, ρδ refers to the spin state that is driven by the microwave with adetuning of. Although details of the lineshape function depend on thedominant broadening mechanisms, it is assumed here that g(δ) is aGaussian-shape function with a full-width-at-half maximum (FWHM)linewidth of δv.

The QRB-Guidestar Feedback. In this section, the QRB-GS feedback isformulated in detail. As detailed herein, ρ=ρ₁⊗ . . . ⊗ρM labels theinitial spin state of QRBs at positions of x₁, . . . , x_(M) inside ofthe complex medium. An external magnetic field gradient separatesindividual resonance frequencies {v_(m)} of {ρ_(m)}(m=1, 2, . . . , M).{E_(m)} resonantly modulate the spin density operators {ρ_(m)} withlineshape functions {g_(m)} and linewidths {δv_(m)}. In principle,{x_(m)} could then be reconstructed from {v_(m)} and knowledge of theexternal magnetic field.

Optical fields inside of the complex medium are described by atransmission matrix T. For example, the matrix element t_(m,n) describesthe optical field at {x_(m)} when the nth incident basis mode couplesinto the medium, i.e., |t_(m,n)|² is the optical intensity that excitesthe QRB at x_(m). The internal optical fields excite the embedded QRBs,which in turn emit spin-dependent broadband fluorescence. By driving thespin resonance of target QRBs, the spin-dependent fluorescence providesinformation of the transmission matrix elements.

Specifically, obtaining the QRB-GS feedback S_(i,n) at x_(i) for the nthincident basis mode proceeds as follow. First, the fluorescence photonsof the initial spin state ρ are collected with the basis mode. Thephoton numbers N_(n) collected for unit integration time is:

$\begin{matrix}{{N_{n}\lbrack\rho\rbrack} = {\sum\limits_{m}^{M}{{t_{m,n}}^{2}{\left\{ {{\gamma_{0}\sigma_{00}^{m}} + {\gamma_{1}\sigma_{11}^{m}}} \right\}.}}}} & (6)\end{matrix}$

Here, γ₀ (γ₁) represents the collected spin-dependent photon numbers of|0

(

1|) for unit excitation intensity. σ₀₀ ^(m) and σ₁₁ ^(m) account for thespin population (i.e. σ₀₀ ^(m)=

0|ρ_(m)|0

, σ₁₁ ^(m)=

1|ρ_(m)|1

)| of the QRBs. It is assumed that the photon collection are identicalfor all embedded QRBs.

Next, a microwave is applied that is resonant to the target ith-QRB andrepeat the fluorescence photon collection. As introduced in Eq. (4) and(5), the microwave operation, which is represented through a quantumoperator E_(i), modulates the spin population by Δσ_(i) for the targetQRB and by g_(m)(δ_(m) ^(i)) Δσ_(m) for the other ‘background’ mth-QRB,where δ_(m) ^(i)=v_(m)−v_(i)(i.e. δ₁ ^(i)=0). Then the collected photonnumber N_(n) is

$\begin{matrix}{{N_{n}\left\lbrack {E_{i}(\rho)} \right\rbrack} = {{{t_{i,n}}^{2}\left( {{\gamma_{0}\left( {\sigma_{00}^{i} - {\Delta\sigma}_{i}} \right)} + {\gamma_{1}\left( {\sigma_{11}^{i} + {\Delta\sigma}_{i}} \right)}} \right\}} + {\sum\limits_{m \neq i}^{M}{{t_{m,n}}^{2}{\left\{ {{\gamma_{0}\left( {\sigma_{00}^{m} - {{g_{m}\left( \delta_{m}^{i} \right)}{\Delta\sigma}_{m}}} \right)} + {\gamma_{1}\left( {\sigma_{11}^{m} + {{g_{m}\left( \delta_{m}^{i} \right)}{\Delta\sigma}_{m}}} \right)}} \right\}.}}}}} & (7)\end{matrix}$

By subtracting Eq. (7) from (6),

$\begin{matrix}{{{N_{n}\lbrack\rho\rbrack} - {N_{n}\left\lbrack {E_{i}(\rho)} \right\rbrack}} = {{{{t_{i,n}}^{2}{{\Delta\sigma}_{i}\left( {\gamma_{0} - \gamma_{1}} \right)}} + {\sum\limits_{m \neq i}^{M}{{t_{m,n}}^{2}{g_{m}\left( \delta_{m}^{i} \right)}{{\Delta\sigma}_{m}\left( {\gamma_{0} - \gamma_{1}} \right)}}}}..}} & (8)\end{matrix}$

If δ_(m) ^(i)≥δv_(m), for all m≠i, the contribution from the backgroundQRBs is ignorable, reducing Eq. (8) to the desirable QRB-GS feedback atx_(i):

S _(i,n) =N _(n)[ρ]−N _(n)[E _(i)(ρ)]=|t _(i,n)|²Δσ_(i)(Γ₀−Γ₁),  (9)

The condition δ_(m) ^(i)≥δv_(m) determines our spatial resolution of theQRB-GS feedback, with an analogous to Rayleigh resolution limit inconventional optical microscopy. For a given external magnetic fieldgradient dB/dx, the ESR lineshape g(δ_(m) ^(i)) is translated to aneffective point spread function (PSF) of the QRB-GS feedback. Thus, ourspatial resolution Δd_(QRB) is given by the FWHM resolution of theeffective PSF:

${\Delta \; d_{QRB}} = {\frac{\delta \; v}{\gamma_{e}\left( {{dB}/{dx}} \right)}.}$

Here, δv is the mean ESR linewidth of QRBs (i.e.,

$\left. {\frac{1}{M}{\sum\limits_{m = 1}^{M}{\delta \; v_{m}}}} \right).$

This spatial resolution can be improved by introducing assumptions suchas Δσ₁=Δσ₂= . . . =Δσ_(M). This resolution improvement depends on theaccuracy of the assumptions and the signal-to-noise ratio of themeasurements as in conventional optical microscopy.

Noise Estimation. The noise in the QRB-GS feedback can be modeled withthe photon shot noise of spin-dependent fluorescence. For the QRB-GSfeedback

S _(i,n) =N _(n)[ρ]−N _(n)[E(ρ)]=|t _(i,n)|²Δσ_(i)(Γ₀−Γ₁),  (9)

consider the photon shot noise in N_(n)[ρ] and N_(n)[E(ρ)]:

$\begin{matrix}{\frac{N_{i,n}}{S_{i,n}} = {\frac{\sum\limits_{m}{{t_{m,n}}\sqrt{1 - {\sigma_{11}^{m}C}}}}{{t_{i,n}}^{2}{\Delta\sigma}_{i}\sqrt{\gamma_{0}}C} \times \left\lbrack {1 + \sqrt{1 - \frac{{\Delta\sigma}_{m}{g_{m}\left( \delta_{m}^{i} \right)}C}{1 - {\sigma_{11}^{m}C}}}} \right\rbrack}} \\{\simeq {\frac{\sum\limits_{m}{{t_{m,n}}\sqrt{1 - {\sigma_{11}^{m}C}}}}{{t_{i,n}}^{2}{\Delta\sigma}_{i}\sqrt{\gamma_{0}}C} \times \left\lbrack {2 - \sqrt{1 - \frac{{\Delta\sigma}_{m}{g_{m}\left( \delta_{m}^{i} \right)}C}{2\left( {1 - {\sigma_{11}^{m}C}} \right)}}} \right\rbrack}} \\{\overset{<}{\sim}\frac{2{\sum\limits_{m}{t_{m,n}}}}{{t_{i,n}}^{2}{\Delta\sigma}_{i}\sqrt{\gamma_{0}}C}} \\{\simeq \frac{2M\sqrt{\left( {a/A} \right)T}}{\left( {a/A} \right)T\; {\Delta\sigma}_{i}\sqrt{\gamma_{0}}C}} \\{= \frac{2M}{\sqrt{\gamma_{0}{aT}}{\Delta\sigma}_{i}C}}\end{matrix}.$

Assuming that the complex medium is in a lossless waveguide whosecross-section area is A, the noise N_(i,n) to signal Eq. (9) ratio is

${t_{i,n}}\sqrt{{\gamma_{0}\sigma_{00}^{i}} + {\gamma_{1}\sigma_{11}^{i}}}$${t_{i,n}}\sqrt{{\gamma_{0}\left( {\sigma_{00}^{i} - {\Delta\sigma}_{i}} \right)} + {\gamma_{1}\left( {\sigma_{11}^{i} + {\Delta\sigma}_{i}} \right)}}$$\sum\limits_{m \neq i}^{M}{{t_{m,n}}\sqrt{{\gamma_{0}\sigma_{00}^{m}} + {\gamma_{1}\sigma_{11}^{m}}}}$$\sum\limits_{m \neq i}^{M}{{t_{m,n}} \times \sqrt{{\gamma_{0}\left\{ {\sigma_{00}^{m} - {{g_{m}\left( \delta_{m}^{i} \right)}{\Delta\sigma}_{m}}} \right\}} + {\gamma_{1}\left\{ {\sigma_{11}^{m} + {{g_{m}\left( \delta_{m}^{i} \right)}{\Delta\sigma}_{m}}} \right\}}}}$

Here, Σ_(m=1) ^(M)|t_(m,n)|²=(a/A)T=aT where a is the cross-sectionalarea of QRBs, C=1−γ₁/γ₀, and T is the total transmission of the complexmedium. As expected, the noise-to-signal ratio approaches to zero for alonger integration.

Four-Phase Method with the QRB-GS Feedback

To access the phase of the transmission matrix element t_(m,n), theincident Fourier basis modes {k_(n)} are encoded into the holographicilluminations by interfering themselves with the reference mode u_(ref).Specifically, the holographic illumination of a incident basis modek_(n) is represented by

E _(n) ^(in)(ϕ)=1+e ^(i(k) ^(n) ^(·r+ϕ)),  (10)

where u_(ref)=1, and ϕ is the phase of the basis mode relative to thereference mode. When E_(n) ^(in)(ϕ) couples to the complex medium, thescattered optical field on the ith QRB is

$\begin{matrix}{{E_{i,n}^{out}(\varphi)} = {{TE}_{n}^{in} = {t_{i,R} + {t_{i,n}e^{i\; \varphi}}}}} \\{= {t_{i,R}\left( {1 + {\frac{t_{i,n}}{t_{,R}}e^{i\; \varphi}}} \right)}} \\{= {1 + {t_{i,n}e^{i\; \varphi}}}}\end{matrix},$

t_(i,R) is substituted for 1 without loss of generality. This leads theQRB-GS feedback S_(i,n)(ϕ):

$\begin{matrix}{{S_{i,n}(\varphi)} = {{{E_{i,n}^{out}(\varphi)}}^{2} \times {{\Delta\sigma}_{i}\left( {\gamma_{0} - \gamma_{1}} \right)}}} \\{= {\left\lbrack {1 + t_{i,n}^{2} + {2{t_{i,n}}{\cos \left( {\varphi + {\arg \left( t_{i,n} \right)}} \right)}}} \right\rbrack \times {{{\Delta\sigma}_{i}\left( {\gamma_{0} - \gamma_{1}} \right)}.}}}\end{matrix}$

By measuring the QRB-GS feedback with the four phase shifts ϕ=0, π/2, π,and 3π/2, the phase of the matrix element is reconstructed by

${\arg \left( t_{i,n} \right)} = {{\arg \left\lbrack {\frac{{S_{i,n}(0)} - {S_{i,n}(\pi)}}{4} + {i\frac{{S_{i,n}\left( {3{\pi/2}} \right)} - {S_{i,n}\left( {\pi/2} \right)}}{4}}} \right\rbrack}.}$

Phase Readout with Continuously-Driven ESR. The phase of thetransmission matrix element is determined by sinusoidally modulating theoptical excitation, as the phase of the incident basis modes is sweptrelative to the reference mode. In the meantime, Δσ withcontinuously-driven ESR depends on the optical excitation, since thesteady-steady solutions ρ and E(ρ) of Eq. (3) are a function of opticalpumping. This dependence produces non-linearity of the four-phase shiftmeasurement with the QRB-GS feedback. In this section, it is shown thatthe small variation of Δσ does not affect on the phase readout in ourmeasurement up to the first order. For the holographic illuminationÊ_(n) ^(in)1+e^(i(k) ^(n) ^(·r+ϕ)), the optical excitation I_(i,n)(ϕ) onthe ith QRB is

I _(i,n)(ϕ)=|E _(i,n) ^(out)(ϕ)|²=1+t _(i,n) ²+2t _(i,n) cos(ϕ+θ_(i,n)),

where arg(t_(i,n)) is substituted to θ_(i,n). The small variation of Δσis introduced up to the first order, while the phase ϕ of theholographic illumination is modulated:

Δσ_(i)(φ) ≃ Δσ_(i)⁽⁰⁾ + Δσ_(i)⁽¹⁾(φ)  where${\Delta\sigma}_{i}^{(0)} = {\left. {\Delta\sigma}_{i} \middle| {}_{I_{i,n}{(0)}}{{- {I_{i,n}(0)}}\frac{d\; {\Delta\sigma}_{i}}{dI}} \middle| {}_{I_{i,n}{(0)}}{{\Delta\sigma}_{i}^{(1)}(\varphi)} \right. = \left. {{I_{i,n}(\varphi)}\frac{d\; {\Delta\sigma}_{i}}{dI}} \middle| {}_{I_{i,n}{(0)}}. \right.}$

The corresponding QRB-GS feedback is

$\begin{matrix}{{S_{i,n}(\varphi)} = {{I_{i,n}(\varphi)}{{\Delta\sigma}_{i}(\varphi)}\left( {\gamma_{0} - \gamma_{1}} \right)}} \\{= {{{I_{i,n}(\varphi)}\left\lbrack {{\Delta\sigma}_{i}^{(0)} + {{\Delta\sigma}_{i}^{(1)}(\varphi)}} \right\rbrack}\left( {\gamma_{0} - \gamma_{1}} \right)}} \\{{= {{S_{i,n}^{(0)}(\varphi)} + {S_{i,n}^{(1)}(\varphi)}}},}\end{matrix}$ where $\begin{matrix}{{S_{i,n}^{(0)}(\varphi)} = {\left( {\gamma_{1} - \gamma_{1}} \right){\Delta\sigma}_{i}^{(0)}{I_{i,n}(\varphi)}}} \\{\doteq {\alpha_{i,n}^{(0)} + {\alpha_{i,n}^{(1)}{\cos \left( {\varphi + \theta_{i,n}} \right)}}}}\end{matrix}$ $\begin{matrix}{{S_{i,n}^{(1)}(\varphi)} = \left. {\left( {\gamma_{0} - \gamma_{1}} \right){I_{i,n}(\varphi)}^{2}\frac{d\; {\Delta\sigma}_{i}}{dI}} \right|_{I_{i,n}{(0)}}} \\{\doteq {\beta_{i,n}^{(0)} + {\beta_{i,n}^{(1)}{\cos \left( {\varphi + \theta_{i,n}} \right)}} + {\beta_{i,n}^{(2)}{{\cos^{2}\left( {\varphi + \theta_{i,n}} \right)}.}}}}\end{matrix}$

Since cos²(θ_(i,n)+π)=cos² θ_(i,n), andcos²(θ_(i,n)+3π/2)=cos²(θ_(i,n)+π/2), the nonlinear dependence in thefour-phase measurement is cancelled out, resulting in

${\arg \left\lbrack {\frac{{S_{i,n}(0)} - {S_{i,n}(\pi)}}{4} + {i\frac{{S_{i,n}\left( {3{\pi/2}} \right)} - {S_{i,n}\left( {\pi/2} \right)}}{4}}} \right\rbrack} = {{\arg \left\lbrack {\frac{\alpha_{i,n}^{(1)} + \beta_{i,n}^{(1)}}{2}\left( {{\cos \; \theta_{i,n}} + {i\; \sin \; \theta_{i,n}}} \right)} \right\rbrack} = {{\arg \left( t_{i,n} \right)}.}}$

Estimation of Spatial Resolution. Described herein is how to estimatethe spatial resolution of achieved subwavelength foci. Here, it isassumed that the target QRB₁ and QRB₂ are point-like particles localizedat x₁ and x₂, respectively. In continuously-driven ESR spectroscopy, thespin density operators ρ₁ and ρ₂ are optically polarized into |0

0| when the microwave frequency v_(off) is far off from their resonancefrequencies, v₁ and v₂. By contrast, when the microwave is on resonanceswith v₁ and v₂, ρ₁ and ρ₂ become (1−Δσ)|0

0|+Δσ|1

1| with Δσ=1/2, provided that the QRBs are not optically saturated.

In this analysis, I₁ ⁽¹⁾ and I₂ ⁽¹⁾ (I₁ ⁽²⁾ and I₂ ⁽²⁾) is denoted asthe optical excitation at x₁ and x₂ when the wavefront W_(v1)(W_(v2)) isprojected. N⁽¹⁾(v) (N⁽²⁾(v)) is the corresponding ODMR spectra withW_(v1)(W_(v2)) projection. To estimate the spatial resolution Δr⁽¹⁾ ofthe subwavelength focus at x₁ with W_(v1) projection, consider therelations,

$\begin{matrix}{{N^{(1)}\left( v_{off} \right)} = {{I_{1}^{(1)}\gamma_{0}} + {I_{2}^{(1)}p\; \gamma_{0}}}} & (11) \\\begin{matrix}{\left. {{N^{(1)}\left( v_{1} \right)} = {I_{1}^{(1)}\left\lbrack {{\gamma_{0}\left( {1 - {\Delta\sigma}} \right)} + {\gamma_{1}{\Delta\sigma}}} \right)}} \right\rbrack + {I_{2}^{(1)}p\; \gamma_{0}}} \\{= {{I_{1}^{(1)}\left( {\gamma_{0} - \frac{\Delta \; \gamma}{2}} \right)} + {I_{2}^{(1)}p\; \gamma_{0}}}}\end{matrix} & (12) \\\begin{matrix}{{N^{(1)}\left( v_{2} \right)} = {{I_{1}^{(1)}\gamma_{0}} + {I_{2}^{(1)}{p\left\lbrack {{\gamma_{0}\left( {1 - {\Delta\sigma}} \right)} + {\gamma_{1}{\Delta\sigma}}} \right\rbrack}}}} \\{= {{I_{1}^{(1)}\gamma_{0}} + {I_{2}^{(1)}p\; {\left( {\gamma_{0} - \frac{\Delta \; \gamma}{2}} \right).}}}}\end{matrix} & (13)\end{matrix}$

Here, Δγ=γ₀−γ₁, and the parameter p takes account of the NV densitydifference between the two QRBs. p can be determined from ODMRN^((c1))(v) under the diffraction-limited excitation, in which theoptical excitations at x₁ and x₂ are approximately equal:

$\begin{matrix}{p = {\frac{{N^{({cl})}\left( v_{off} \right)} - {N^{({cl})}\left( v_{2} \right)}}{{N^{({cl})}\left( v_{off} \right)} - {N^{({cl})}\left( v_{1} \right)}}.}} & (14)\end{matrix}$

In this experiment, p is determined with W_(c1) projection.

From the ODMR spectra plotted in FIG. 4D the ODMR is obtained atv_(off), v₁, and v₂ by fitting to the Lorentzian lineshape function:

N ⁽¹⁾(v ₁)/N ⁽¹⁾(v _(off))=0.9902(0.9891, 0.9912)

N ⁽¹⁾(v ₂)/N ⁽¹⁾(v _(off))=0.9988(0.9976, 0.9999)

N ⁽²⁾(v ₁)/N ⁽²⁾(v _(off))=0.9994(0.9985, 1.0004)

N ⁽²⁾(v ₂)/N ⁽²⁾(v _(off))=0.9878(0.9870, 0.9885)

N ^((c1))(v ₁)/N ^((c1))(v _(off))=0.9954(0.9943, 0.9965)

N ^((c1))(v ₂)/N ^((c1))(v _(off))=0.9942(0.9932, 0.9952)

where the values in the parenthesis represent 95% confidence bound ofthe fitting. By inserting the fit values to Eq. (11-14), it was found I₁⁽¹⁾/I₂ ⁽¹⁾≃10.1 and p≈1.261. Similarly, I₂ ⁽²⁾/I₁ ⁽²⁾≃17.1.Assumingsubwavelength focus features a Gaussian intensity shape, its spatialresolution Δr⁽¹⁾ and Δr⁽²⁾ are the FWHM of the intensity shapes:

${\Delta \; r^{(1)}} = {{\sqrt[2]{\frac{\ln \; 2}{\ln \left( {I_{1}^{(1)}/I_{2}^{(1)}} \right)}}\Delta \; x} \simeq {204\mspace{14mu} {nm}}}$${{\Delta \; r^{(2)}} = {{\sqrt[2]{\frac{\ln \; 2}{\ln \left( {I_{2}^{(2)}/I_{1}^{(2)}} \right)}}\Delta \; x} \simeq {184\mspace{14mu} {nm}}}},$

where Δx=|x₁−x₂|=186 nm (FIGS. 7A-7D). The uncertainties of theestimations are plotted in FIG. 4E.

FIG. 8 illustrates a method 800 for optical focusing, according toembodiments. Aspects of the method can encompass the approaches outlinesfor FIGS. 1-7 above and can be executed at least in part by the systems300, 900 as illustrated in FIGS. 3A-3C, FIG. 9, respectively.

The method 800 can include, at 810, applying a magnetic field to asample that exhibits optical scattering. The sample can have multipleemission particles (QRBs) distributed therein that exhibitspin-dependent fluorescence. The magnetic field shifts a resonancefrequency of each emission particle in a position-dependent manner. Insome embodiments, the magnetic field has a gradient. In someembodiments, the magnetic field has a constant magnitude (e.g., constantpeak magnitude) as well as a constant gradient, such that the magneticfield shifts the resonance frequency of each emission particle based onboth its position and orientation.

The method 800 can further includes estimating the position and/ororientation of the emission particle(s) along two or more axes. Forexample, as also illustrated in FIG. 3D for a single axis, the magneticfield can be first applied along a first axis of the sample (e.g., anX-axis, for simplicity of explanation), and the position/orientation ofthe particle along that axis is estimated. This step can then berepeated along a second axis different from the first axis (e.g., aY-axis, that can be orthogonal to the first axis), along a third axis(e.g., a Z-axis, that can be orthogonal to the first axis and the secondaxis), and so on, to estimate a position/orientation of the particles inthree dimensions.

The sample can include an optical super-lens coupled thereto, i.e., alens embedded with a material useful for extending the resolution of thelens beyond its diffraction limit. Such materials, sometimes alsoreferred to as metamaterials, can include (but are not limited to) gold,silver, and/or the like. In some embodiments, the super-lens is a layeron the sample and interfaces an excitation objective to control deliveryof an excitation beam to the sample. Additionally or alternatively, thesuper-lens could be part of the imaging system itself, e.g., the system300 and/or the system 900. In some embodiments, each emission particlewithin the sample includes a solid-state quantum emitter such as, forexample, a diamond crystal (e.g., bulk diamond, a diamond nanocrystal,and/or the like) having a nitrogen vacancy color center, or a setthereof. FIG. 10 illustrates an example placement of a super-lens 1013as coupled to the sample 1010 having emission particles 1015 (e.g.,nanodiamonds) embedded therein. The objective 1012 b (e.g., similar tothe objective 312) then receives the fluorescence emission from theparticles 1015 via the super-lens 1013 as illustrates.

The method 800 also includes, at 820, exciting the sample with anexcitation beam, such that the excitation beam causes at least one ofthe emission particles to emit spin-dependent fluorescence, as generallyillustrated in FIG. 1B. The excitation beam can include one or morebasis modes, as generally illustrated as pixels in FIG. 1A. The samplecan be excited from multiple angles of incidence such as by, forexample, use of one or more scanning mirrors, acousto-optic deflectors,and/or the like. The excitation beam can include a reference part and aset of basis modes.

In some embodiments, the method 800 may also include applying, to theemission particle(s), a microwave signal to modulate a spin state (e.g.,the spin state ρ, as discussed herein for FIGS. 1-7) of at least one ofthe emission particles.

The method 800 also includes, at 830, detecting the spin-dependentfluorescence emitted by at least one of the emission particles.

The method 800 also includes, at 840, estimating a position of theemission particle(s) within the sample based on a) the spin-dependentfluorescence, b) the resonance frequency of the emission particle(s),and c) the magnetic field.

The method 800 also includes, at 850, estimating optical transmissioninformation for the sample based on a wavefront of the excitation beamand the estimated position. The optical transmission informationincludes a measure of an optical field at each position of an emissionparticle of the plurality of emission particles.

In some embodiments, when the sample is illuminated from multipleincidence angles, the optical transmission information can be furtherbased on/account for the use of such multiple incidences of theexcitation beam. In some embodiments, when the excitation beam includesa reference part, a phase of an optical transmission coefficient of theoptical transmission information can be determined by, for example,using optical interferometry.

In some embodiments, when the excitation beam includes one or more basismodes (e.g., a set of basis modes), the optical transmission informationcan include a measure of the optical field for each such basis mode. Insuch embodiments, the method 800 can further include estimating, for theposition of each emission particle and for each basis mode, a feedbackparameter to generate a set of feedback parameters. In some embodiments,each feedback parameter is estimated for each emission particle and foreach basis mode generally as follows. First, a fluorescence signal(e.g., a first fluorescence signal) is detected as laid out herein forstep 830. Then, the microwave signal is applied to the sample with afrequency that is resonant with a frequency of that emission particle.Then, a second fluorescence signal is detected from the sample inresponse to the excitation beam and the microwave signal, and thefeedback parameter can be estimated for that emission particle and thatbasis mode based on the second fluorescence signal.

In some additional or alternative embodiments, the feedback parametercan be estimated (again, for each emission particle and each basis mode)based on the portion of the excitation beam that is reflected and/ortransmitted through the sample, rather than on fluorescence. Generally,the amount of fluorescence from the sample is proportional to the amountof absorption of the excitation beam. Modulating spin-dependentfluorescence with the microwave signal modulates the absorption of theexcitation beam, which can be measured by recording the reflected and/ortransmitted excitation beam. This modulation of the reflected and/ortransmitted excitation beam can then be employed to produce the feedbackparameter. Since the reflected and/or transmitted excitation beams arecoherent, they can contain additional information that can be decoded.

First, the reflected and/or transmitted portion of the excitation beamis detected, such as via an imaging device, as a first excitation beamsignal. Then, the microwave signal is applied to the sample with afrequency that is resonant with a frequency of that emission particle.Then, the reflected and/or transmitted portion of the excitation beam isagain detected, this time as a second excitation beam signal, and thefeedback parameter can be estimated for that emission particle and thatbasis mode based on the second excitation beam signal.

This set of feedback parameters can then be used to modify acharacteristic of the excitation beam. For example, in some embodiments,the amplitude and/or phase associated with one or more basis modes canbe modified using a spatial light modulator.

FIG. 9 is a schematic illustration of an environment/system 900 in whichoptical focusing in a sample 910 including emission particles 915 (e.g.,similar to the nanodiamonds illustrated in FIG. 3A) within the sample asdisclosed herein may be implemented and/or carried out. In someembodiments, aspects of the system 900 can be structurally and/orfunctionally similar to the systems, apparatuses, and/or devicesdescribed herein with respect to FIGS. 3A-3C, and/or can perform atleast part of the method described in FIG. 8.

The system 900 can interface with the sample 910 including the emissionparticles 915 in any suitable manner, and (as indicated here by dottedlines) the sample 910 need not constitute part of the system 900. Thesystem 900 includes a magnet 920, a microwave generator 930, a lightsource 940, a wavefront shaping device 950 in optical communication withthe light source, a detector 960, and a controller 980. In someembodiments, the system 900 (as illustrated) can also include an imagingdevice 970 and a memory 990. Unless explicitly noted otherwise, any ofthese components may be in communication with each other in any suitablemanner (e.g., optical, electronic, electrical, and/or the like).

In some embodiments, all components of the system 900 can be included ina common casing such as, for example, a single housing that presents thesystem 900 as an integrated, one-piece apparatus for a user. In otherembodiments, at least some components of the system 2300 can be inseparate locations, housings, and/or apparatuses. For example, in someembodiments, the controller 980 and the memory 990 can be included in aseparate compute device can be a smartphone in communication with theother components via one or more networks, each of which can be any typeof network such as, for example, a local area network (LAN), a wide areanetwork (WAN), a virtual network, a telecommunications network, and/orthe Internet, implemented as a wired network and/or a wireless network.Any or all communications can be secured (e.g., encrypted) or unsecured,as is known in the art. The system 900 can be or encompass a personalcomputer, a server, a work station, a tablet, a mobile device, a cloudcomputing environment, an application or a module running on any ofthese platforms, and/or the like.

The system 900 can also encompass a database (not shown), although itwill be understood that, in some embodiments, the database and thememory 990 can be a common data store. The system 900 can also includeone or more input/output (I/O) interfaces (not shown), implemented insoftware and/or hardware, for other components of the system 900, and/orexternal to the system 900, to interact with the system.

The memory 990 and/or the database can independently be, for example, arandom access memory (RAM), a memory buffer, a hard drive, a database,an erasable programmable read-only memory (EPROM), an electricallyerasable read-only memory (EEPROM), a read-only memory (ROM), Flashmemory, and/or so forth. The memory 990 and/or the database can storeinstructions to cause the controller 980 to execute processes and/orfunctions associated with the system 900 such as, for example, one ormore steps of the method 800.

The controller 980 can be any suitable processing device configured torun and/or execute a set of instructions or code associated with thesystem 900. The controller 980 can be, for example, a speciallyprogrammed processor, Field Programmable Gate Array (FPGA), ApplicationSpecific Integrated Circuit (ASIC), or Digital Signal Processor (DSP).

Explained with reference to a typical use scenario, the sample 910 canbe a scattering sample and the emission particles 915 embedded thereincan exhibit spin-dependent fluorescence. In some embodiments, eachemission particle can include a solid-state quantum emitter such as, forexample, a diamond nanocrystal having a nitrogen vacancy center.

The magnet 920 can apply, to the sample 910, a magnetic field thatshifts a resonance frequency of each emission particle 915 in aposition-dependent manner. In some embodiments, the magnetic field has agradient, that results in the resonance frequency shift of the emissionparticles 915. In some embodiments, the magnetic field has a constantmagnitude (e.g., a constant peak magnitude) as well as a constantgradient that results in the resonance frequency shift of the particles915 based on both their position as well as orientation. In someembodiments, the magnet 920 and/or the sample 910 are movable withrespect to each other, such that the magnet 920 (e.g., when it includes2 or more electromagnets) can apply the gradient magnetic field alongtwo or more axis of the sample to ascertain the position/orientation ofthe emission particles 915 in two or more dimensions. As noted hereinwith respect to Eq. 2 and generally described with respect to FIG. 3A,in some cases the QRBs can have different orientation, and the magnet920 can be used to generate a constant field that can be used to probethe QRBs via Zeeman splitting and direct fluorescence imaging

For example, the magnet can apply the magnetic field sequentially alonga first, second, third axis (e.g., an X, Y, Z axis, respectively) toaccurately ascertain the position/orientation of each emission particle915 in space, and within the sample 910. In some embodiments, the magnet920 can include a permanent magnet such as that including magneticmetallic elements, rare-earth magnets, and/or the like. In someembodiments, the magnet 920 can include an electromagnet controllable bythe controller 980.

In a non-limiting example, the magnet 920 encompasses a set of threeelectromagnets generating magnetic fields B_(x)(r)=(x,0,0),B_(y)(r)=(0,x,0), and B_(z)(r)=(0,0,x). In some cases, one of themagnetic field gradients can be suitably increased as needed, such asdue to sample degeneracy (e.g., B_(x)(r)=(10x,0,0)). The electromagnetgenerating B_(x)(r) is turned on, and the ODMR spectrum is measured,which gives a set of resonance frequencies {v_(x)(x)}. Similarly,B_(y)(r) and B_(z)(r) are turned on sequentially, the corresponding ODMRspectra are measured and consequently sets of resonance frequencies{v_(y)(x)} and {v_(z)(x)}, respectively are determined. To determine theposition x from v_(x)(x), v_(y)(x), v_(z)(x), the following equationscan be solved:

γ_(e)×{x}×sin[{θ(x)}]×cos[{φ(x)}]={v _(x)(x)}

γ_(e)×{x}×sin[{θ(x)}]×sin[{φ(x)}]={v _(y)(x)}

γ_(e)×{x}×cos[{θ(x)}]={v_(z)(x)}

The following equation can also be considered in evaluating the positionx:

γ_(e)×{x}=({v _(x)(x)}² +{v _(y)(x)}² +{v _(z)(x)}²)^(1/2)

The microwave generator 930, which can be similar to the microwavesource in FIG. 3B, can generate and apply a microwave signal to thesample 910, which in turn modulates the spin state of at least one ofthe emission particles 915. The light source 940 and the wavefrontshaping device 950 collective generate and apply an excitation beam toexcite the sample, which results in spin-dependent fluorescence from atleast one of the emission particles. The excitation beam can include aset of basis modes (e.g., shown as individual pixels in FIG. 1A), andthe wavefront shaping device 950 can be designed to manipulate/modifyone or more characteristics (e.g., amplitude, phase) of each basis modeseparately. In some embodiments, the light source 940 can excite thesample from different angles of incidence such as, for example, by usingone or more scanning mirrors, acousto-optic deflectors, and/or the liketo modify incidence of the excitation beam on the sample 910.

The light source 940 can be any suitable light source to affectexcitation of the emission particles 915, such as, for example, the 532nm laser illustrated in FIG. 3B, a light source emitting in the 532-635nm wavelength range, etc., and may depend on the fluorescencecharacteristics of the emission particles 915. The wavefront shapingdevice 950 can include any spatial light modulator such as, but is notlimited to, a digital micromirror device (DMD, as illustrated in FIG.3B), a liquid crystal-on-silicon spatial light modulator, amagneto-optic spatial light modulator, and a deformable spatial lightmodulator.

The detector 960 detects the spin-dependent fluorescence of the emissionparticles 915 emanating from the sample 910. In some embodiments, thedetector 960 is a spectroscopic detector such as, for example, asingle-photon counting module (SPCM) as illustrated in FIG. 3B.

The controller 980, which is communicably coupled to the detector 960,can then estimate/compute the position of each emission particle 915within the sample based on a combination of the spin dependentfluorescence, the (known) resonant frequency of the emission particles915, and the magnetic field. As generally explained with reference toFIGS. 1-7, this estimation of the position can then be employed toestimate optical transmission properties (e.g., the optical transmissionmatrix T) of the scattering sample 910; accordingly, the controller 980can estimate optical transmission information for the sample based onthe wavefront of the excitation beam and the estimation positions of theemission particles 915. This optical transmission information caninclude a measure of optical field at each position of an emissionparticle 915 within the sample 910. In some embodiments, the opticaltransmission information can be specific for each basis mode of theexcitation beam, i.e., it can include a measure of optical field foreach basis mode. In some embodiments, when the light source 920 canexcite the sample via multiple angles of incidence, the opticaltransmission information can account for, and/or otherwise be based on,the differing incident angles of the excitation beam, and particularlyof the differing incident angles of the basis modes of the excitationbeam.

The controller 980 can further estimate one or more feedback parameters(e.g., the QRB-GS feedback) that can be applied to the wavefront shapingdevice 950 to manipulate the wavefront of the excitation beam forsub-resolution focusing into the sample 910. When done iteratively, thewavefront of the excitation beam can be optimized for imaging the sample910.

In some embodiments, the controller 980 can estimate the feedbackparameter for each emission particle and each basis mode, generally asfollows. First, the fluorescence signal (e.g., a first fluorescencesignal) is detected as disclosed herein. Then, after the microwavegenerator 930 applies the microwave signal to the sample 910, a secondfluorescence signal is acquired by the detector 960, which can then alsobe employed to estimate that feedback parameter.

In some additional or alternative embodiments, the controller 980 canestimate the feedback parameter (again, for each emission particle andeach basis mode) based on the portion of the excitation beam that isreflected and/or transmitted through the sample, rather than onfluorescence, generally as follows. First, the reflected and/ortransmitted wavefront of the excitation beam is detected via the imagingdevice 970 as a first excitation beam signal. Then, the microwave signalis applied to the sample with a frequency that is resonant with afrequency of that emission particle. Then, the reflected and/ortransmitted wavefront of the excitation beam is again detected, thistime as a second excitation beam signal, and the feedback parameter canbe estimated for that emission particle and that basis mode based on thesecond excitation beam signal. The wavefront shaping device 950 can,based on the feedback parameter, modify at least one characteristic(e.g., phase, amplitude) of the wavefront of the excitation beam forthat basis mode. In some embodiments, the controller 980 controlsactuation of the elements of the wavefront shaping device 950 to affectwavefront modification.

As generally disclosed herein with respect to FIGS. 1-7, in someembodiments, the system 900 can encompass a microscopy-based imagingsystem that includes an objective lens (as best illustrated by the 100 xobjective lens in FIG. 3B) coupled to the sample 910 and having anumerical aperture NA. The resolution of conventional imaging systemsincluding such objectives is λ/(2*NA) at best, with a, being a centerwavelength of fluorescence of the emission particles 915. Aspects of thefeedback-based modulation of the excitation wavefront as disclosedherein, however, permit sub-resolution imaging below the λ/(2*NA) limit,providing improved resolution in samples that otherwise significantlyscatter any fluorescence emanating from within.

Conclusion

While various inventive embodiments have been described and illustratedherein, those of ordinary skill in the art will readily envision avariety of other means and/or structures for performing the functionand/or obtaining the results and/or one or more of the advantagesdescribed herein, and each of such variations and/or modifications isdeemed to be within the scope of the inventive embodiments describedherein. More generally, those skilled in the art will readily appreciatethat all parameters, dimensions, materials, and configurations describedherein are meant to be exemplary and that the actual parameters,dimensions, materials, and/or configurations will depend upon thespecific application or applications for which the inventive teachingsis/are used. Those skilled in the art will recognize, or be able toascertain using no more than routine experimentation, many equivalentsto the specific inventive embodiments described herein. It is,therefore, to be understood that the foregoing embodiments are presentedby way of example only and that, within the scope of the appended claimsand equivalents thereto, inventive embodiments may be practicedotherwise than as specifically described and claimed. Inventiveembodiments of the present disclosure are directed to each individualfeature, system, article, material, kit, and/or method described herein.In addition, any combination of two or more such features, systems,articles, materials, kits, and/or methods, if such features, systems,articles, materials, kits, and/or methods are not mutually inconsistent,is included within the inventive scope of the present disclosure.

The above-described embodiments can be implemented in any of numerousways. For example, embodiments disclosed herein may be implemented usinghardware, software or a combination thereof. When implemented insoftware, the software code can be executed on any suitable processor orcollection of processors, whether provided in a single computer ordistributed among multiple computers.

The various methods or processes outlined herein may be coded assoftware that is executable on one or more processors that employ anyone of a variety of operating systems or platforms. Additionally, suchsoftware may be written using any of a number of suitable programminglanguages and/or programming or scripting tools, and also may becompiled as executable machine language code or intermediate code thatis executed on a framework or virtual machine.

Also, various inventive concepts may be embodied as one or more methods,of which an example has been provided. The acts performed as part of themethod may be ordered in any suitable way. Accordingly, embodiments maybe constructed in which acts are performed in an order different thanillustrated, which may include performing some acts simultaneously, eventhough shown as sequential acts in illustrative embodiments.

All publications, patent applications, patents, and other referencesmentioned herein are incorporated by reference in their entirety.

All definitions, as defined and used herein, should be understood tocontrol over dictionary definitions, definitions in documentsincorporated by reference, and/or ordinary meanings of the definedterms.

The indefinite articles “a” and “an,” as used herein in thespecification and in the claims, unless clearly indicated to thecontrary, should be understood to mean “at least one.”

The phrase “and/or,” as used herein in the specification and in theclaims, should be understood to mean “either or both” of the elements soconjoined, i.e., elements that are conjunctively present in some casesand disjunctively present in other cases. Multiple elements listed with“and/or” should be construed in the same fashion, i.e., “one or more” ofthe elements so conjoined. Other elements may optionally be presentother than the elements specifically identified by the “and/or” clause,whether related or unrelated to those elements specifically identified.Thus, as a non-limiting example, a reference to “A and/or B”, when usedin conjunction with open-ended language such as “comprising” can refer,in one embodiment, to A only (optionally including elements other thanB); in another embodiment, to B only (optionally including elementsother than A); in yet another embodiment, to both A and B (optionallyincluding other elements); etc.

As used herein in the specification and in the claims, “or” should beunderstood to have the same meaning as “and/or” as defined above. Forexample, when separating items in a list, “or” or “and/or” shall beinterpreted as being inclusive, i.e., the inclusion of at least one, butalso including more than one, of a number or list of elements, and,optionally, additional unlisted items. Only terms clearly indicated tothe contrary, such as “only one of” or “exactly one of,” or, when usedin the claims, “consisting of,” will refer to the inclusion of exactlyone element of a number or list of elements. In general, the term “or”as used herein shall only be interpreted as indicating exclusivealternatives (i.e. “one or the other but not both”) when preceded byterms of exclusivity, such as “either,” “one of,” “only one of,” or“exactly one of. ” “Consisting essentially of,” when used in the claims,shall have its ordinary meaning as used in the field of patent law.

As used herein in the specification and in the claims, the phrase “atleast one,” in reference to a list of one or more elements, should beunderstood to mean at least one element selected from any one or more ofthe elements in the list of elements, but not necessarily including atleast one of each and every element specifically listed within the listof elements and not excluding any combinations of elements in the listof elements. This definition also allows that elements may optionally bepresent other than the elements specifically identified within the listof elements to which the phrase “at least one” refers, whether relatedor unrelated to those elements specifically identified. Thus, as anon-limiting example, “at least one of A and B” (or, equivalently, “atleast one of A or B,” or, equivalently “at least one of A and/or B”) canrefer, in one embodiment, to at least one, optionally including morethan one, A, with no B present (and optionally including elements otherthan B); in another embodiment, to at least one, optionally includingmore than one, B, with no A present (and optionally including elementsother than A); in yet another embodiment, to at least one, optionallyincluding more than one, A, and at least one, optionally including morethan one, B (and optionally including other elements); etc.

In the claims, as well as in the specification above, all transitionalphrases such as “comprising,” “including,” “carrying,” “having,”“containing,” “involving,” “holding,” “composed of,” and the like are tobe understood to be open-ended, i.e., to mean including but not limitedto. Only the transitional phrases “consisting of” and “consistingessentially of” shall be closed or semi-closed transitional phrases,respectively, as set forth in the United States Patent Office Manual ofPatent Examining Procedures, Section 2111.03.

1. A method, comprising: applying, to a sample exhibiting opticalscattering and having a plurality of emission particles distributedtherein that exhibit spin-dependent fluorescence, a magnetic field toshift a resonance frequency of each emission particle of the pluralityof emission particles in a position-dependent manner; exciting thesample with an excitation beam, such that the excitation beam causes atleast one of the plurality of emission particles to emit spin-dependentfluorescence; detecting the spin-dependent fluorescence emitted by atleast one of the plurality of emission particles; estimating a positionof the at least one of the plurality of emission particles within thesample based on the spin-dependent fluorescence, the resonance frequencyof the at least one of the plurality of emission particles, and themagnetic field; and estimating optical transmission information for thesample based on a wavefront of the excitation beam and the estimatedposition, the optical transmission information including a measure of anoptical field at each position of an emission particle of the pluralityof emission particles.
 2. The method of claim 1, further comprisingapplying, to the plurality of emission particles, a microwave signal tomodulate a spin state of at least one of the plurality of emissionparticles.
 3. The method of claim 1, wherein each emission particle ofthe plurality of emission particles includes a solid-state quantumemitter.
 4. The method of claim 3, wherein the solid-state quantumemitter is a diamond nanocrystal having a nitrogen vacancy color center.5. The method of claim 1, wherein the magnetic field has a gradient,such that the applying the magnetic field shifts the resonance frequencyof each emission particle based on the position of that emissionparticle within the sample.
 6. The method of claim 1, wherein themagnetic field has a constant magnitude or a constant gradient, andwherein the applying the magnetic field shifts the resonance frequencyof each emission particle of the plurality of emission particles basedon the position and an orientation of that emission particle within thesample.
 7. The method of claim 1, the applying the magnetic fieldfurther including: applying the magnetic field along a first axis of thesample, such that the estimating the position includes estimating theposition of that emission particle along the first axis; applying themagnetic field along a second axis of the sample, the second axisorthogonal to the first axis, such that the estimating the positionincludes estimating the position of that emission particle along thesecond axis; and applying the magnetic field along a third axis of thesample, the third axis orthogonal to the first axis and to the secondaxis, such that the estimating the position includes estimating theposition of that emission particle along the third axis.
 8. The methodof claim 1, the exciting the sample including exciting the sample from aplurality of angles of incidence with the excitation beam, theestimating the optical transmission information further based on theplurality of angles of incidence.
 9. The method of claim 8, furthercomprising: applying a reference excitation beam to the sample todetermine a phase of an optical transmission coefficient of the opticaltransmission information using one or more optical interferometrictechniques.
 10. The method of claim 1, where the excitation beamincludes a set of basis modes, and wherein the optical transmissioninformation including the measure of optical field for each basis modeof the set of basis modes.
 11. The method of claim 10, furthercomprising: estimating, for the position of each emission particle andfor each basis mode, a feedback parameter to generate a set of feedbackparameters; modifying at least one characteristic of the excitation beambased on the set of feedback parameters.
 12. The method of claim 11,wherein the fluorescence is a first fluorescence signal, the estimatingthe feedback parameter further including, for each emission particle andeach basis mode: applying a microwave signal to the sample, a frequencyof the microwave sample being resonant with the resonance frequency ofthat emission particle; detecting a second fluorescence signal emittedby the sample in response to the excitation beam and the microwavesignal; and estimating the feedback parameter for that emission particleand for that basis mode based on the second fluorescence signal.
 13. Themethod of claim 11, further comprising detecting at least a portion ofthe excitation beam reflected from or transmitted through the sample asa first excitation beam signal, the estimating the feedback parameterfurther including, for each emission particle and each basis mode:applying a microwave signal to the sample, a frequency of the microwavesample being resonant with the resonance frequency of that emissionparticle; detecting a second excitation beam signal reflected from ortransmitted through the sample in response to the excitation beam andthe microwave signal; and estimating the feedback parameter for thatemission particle and for that basis mode based on the second excitationbeam signal.
 14. The method of claim 12, the modifying furthercomprising modifying at least one characteristic of at least one basismode of the excitation beam based on the set of feedback parameters. 15.A system, comprising: a magnet to apply, to a sample exhibiting opticalscattering and having a plurality of emission particles distributedtherein that exhibit spin-dependent fluorescence, a magnetic field toshift a resonance frequency of each emission particle of the pluralityof emission particles in a position-dependent manner; a microwavegenerator to generate and apply a microwave signal to the sample tomodulate the spin state of at least one emission particle of theplurality of emission particles; a light source to excite the samplewith an excitation beam, such that the excitation beam causes at leastone emission particle of the plurality of emission particles to emitspin-dependent fluorescence; a detector to detect the spin-dependentfluorescence; and a controller to: estimate a position of the at leastone of the plurality of emission particles within the sample based onthe spin-dependent fluorescence, the resonance frequency of the at leastone of the plurality of emission particles, and the magnetic field; andestimate optical transmission information for the sample based on awavefront of the excitation beam and the estimated position, the opticaltransmission information including a measure of an optical field at eachposition of an emission particle of the plurality of emission particles.16. The system of claim 15, wherein each emission particle of theplurality of emission particles includes a solid-state quantum emitter.17. The system of claim 16, wherein the solid-state quantum emitter is adiamond nanocrystal having a nitrogen vacancy center.
 18. The system of15, wherein the magnetic field has a gradient to shift the resonancefrequency of each emission particle based on the position of thatemission particle in the sample.
 19. The system of 15, wherein themagnetic field has a constant magnitude or a constant gradient to shiftthe resonance frequency of each emission particle based on the positionand an orientation of that emission particle in the sample.
 20. Thesystem of claim 15, wherein the excitation beam includes a set of basismodes, and wherein the optical transmission information includes themeasure of optical field for each basis mode of the set of basis modes.21. The system of claim 20, wherein the controller estimates, for theposition of each emission particle and for each basis mode, a feedbackparameter to generate a set of feedback parameters.
 22. The system ofclaim 21, wherein the fluorescence is a first fluorescence signal,wherein the controller estimates the feedback parameter by, for the eachemission particle and the each basis mode: controlling the microwavegenerator to apply a microwave signal to the sample, a frequency of themicrowave sample being resonant with the resonance frequency of thatemission particle; controlling the detector to detect a secondfluorescence signal emitted by the sample in response to the excitationbeam and the microwave signal; and estimating the feedback parameter forthat emission particle and for that basis mode based on the secondfluorescence signal.
 23. The system of claim 21, further comprising animaging device for detecting at least a portion of the excitation beamreflected from or transmitted through the sample as a first excitationbeam signal, wherein the controller estimates the feedback parameter by,for the each emission each basis mode: controlling the microwavegenerator to apply the microwave signal to the sample, a frequency ofthe microwave sample being resonant with the resonance frequency of thatemission particle; controlling the imaging device to detect a secondexcitation beam signal reflected from or transmitted through the samplein response to the excitation beam and the microwave signal; andestimating the feedback parameter for that emission particle and forthat basis mode based on the second excitation beam signal.
 24. Thesystem of claim 21, further comprising a wavefront shaping device toreceive the excitation beam and to modify at least one characteristic ofat least one basis mode of the excitation beam based on the set offeedback parameters.
 25. The system of claim 15, further comprising anobjective lens coupled to the sample and having a numerical aperture NA,wherein the fluorescence has a center wavelength λ, and wherein thecontroller and the second detector generate an image of the fluorescenceat a spatial resolution that is less than λ/(2*NA).
 26. The system ofclaim 15, wherein the magnet applies the magnetic field by: applying themagnetic field along a first axis of the sample, such that thecontroller estimates the position by estimating the position of thatemission particle along the first axis; applying the magnetic fieldalong a second axis of the sample, the second axis orthogonal to thefirst axis, such that the controller estimates the position byestimating the position of that emission particle along the second axis;and applying the magnetic field along a third axis of the sample, thethird axis orthogonal to the first axis and to the second axis, suchthat the controller estimates the position by estimating the position ofthat emission particle along the third axis.
 27. The system of claim 15,wherein the light source excites the sample by exciting the sample froma plurality of angles of incidence of the excitation beam, and whereinthe controller estimates the optical transmission information furtherbased on the plurality of angles of incidence.
 28. The system of claim27, further comprising one or more scanning mirrors to modify incidenceof the excitation beam on the sample to effect excitation of the samplefrom the plurality of angles of incidence of the excitation beam. 29.The system of claim 27, further comprising one or more acousto-opticdeflectors to modify incidence of the excitation beam on the sample toeffect excitation of the sample from the plurality of angles ofincidence of the excitation beam.
 30. A system, comprising: a magnet, inelectromagnetic communication with a sample exhibiting opticalscattering and having a plurality of diamond nanoparticles distributedtherein with nitrogen vacancy centers that exhibit spin-dependentfluorescence, to apply a magnetic field having a gradient, such that themagnetic field shifts a resonance frequency of each nitrogen vacancycenter in the plurality of diamond nanoparticles based on a position ofthat nitrogen vacancy center in the sample; a light source to emit anexcitation beam; a wavefront shaping device, in optical communicationwith the light source and the sample, to excite the sample with theexcitation beam, the excitation beam including a set of basis modes,such that the excitation beam causes at least one nitrogen vacancycenter in the plurality of diamond nanoparticles to emit spin-dependentfluorescence; a detector, in optical communication with the sample, todetect the spin-dependent fluorescence emitted by the sample; and acontroller, operably coupled to the detector and the wavefront shapingdevice, to: estimate a position of the at least one nitrogen vacancycenter in the plurality of diamond nanoparticles within the sample basedon the spin-dependent fluorescence, the resonance frequency of the atleast one nitrogen vacancy center in the plurality of diamondnanoparticles, and the magnetic field; generating an opticaltransmission matrix for the sample based on a wavefront of theexcitation beam and the estimated position, the optical transmissionmatrix including a measure of an optical field at a position of eachnitrogen vacancy center in the plurality of diamond nanoparticles, andfor each basis mode of the set of basis modes; estimate, for theposition of each nitrogen vacancy center in the plurality of diamondnanoparticles and for each basis mode, a feedback parameter to generatea set of feedback parameters; and actuate the wavefront shaping deviceto modify at least one characteristic of at least one basis mode of theexcitation beam based on the set of feedback parameters.